I showed how the positive powers of ten and two are interleaved, and said that the interleaving of the negative powers of ten and two is its mirror image. In this article, I will show you why, and prove that the same properties hold.
Powers of Ten and Two in Logarithmic Scale (Nonpositive Powers Highlighted)
To truly understand decimal to binary and binary to decimal conversion, you should know how the powers of ten and the powers of two relate. In particular, you should know how they are interleaved. The interleaving explains why, for example, the number of bits required to represent an n-digit decimal integer varies. Consequently, it also explains why 9,007,199,254,740,991 (253 – 1) is representable in binary floating-point, and why 9,007,199,254,740,993 (253 + 1) is not.
Powers of Ten and Two in Logarithmic Scale (Nonnegative Powers Highlighted)
In this article, I will discuss the interleaving of the positive powers of ten and two, and prove some properties of it. (The interleaving of the negative powers is the mirror image of the positive powers, centered around 100 = 20 = 1.)
In my article “Composing Powers of Two Using The Laws of Exponents” I showed how to combine powers of two using the standard laws of exponents. There are two other rules I use when combining powers of two; I call them the add duplicate power of two rule and the subtract half power of two rule. These are nonstandard rules, applying only to powers of two. Although these are special cases of the existing multiplication and division rules, I’ve found value in recognizing them in addition and subtraction form. I’ll state these rules and show examples of their usage.
I’ve discussed how David Gay’s strtod() function computes an initial floating-point approximation and then uses a loop to check and correct it, but I have not discussed how the correction is made. That is the last piece of the strtod() puzzle, and I will cover it in this article.
In my article “Using Integers to Check a Floating-Point Approximation,” I briefly mentioned “bigcomp,” an optimization strtod() uses to reduce big integer overhead when checking long decimal inputs. bigcomp does a floating-point to decimal conversion — right in the middle of a decimal to floating-point conversion mind you — to generate the decimal expansion of the number halfway between two target floating-point numbers. This decimal expansion is compared to the input decimal string, and the result of the comparison dictates which of the two target numbers is the correctly rounded result.
In this article, I’ll explain how bigcomp works, and when it applies. Also, I’ll talk briefly about its performance; my informal testing shows that, under the default setting, bigcomp actually worsens performance for some inputs.
For decimal inputs that don’t qualify for fast path conversion, David Gay’s strtod() function does three things: first, it uses IEEE double-precision floating-point arithmetic to calculate an approximation to the correct result; next, it uses arbitrary-precision integer arithmetic (AKA big integers) to check if the approximation is correct; finally, it adjusts the approximation, if necessary. In this article, I’ll explain the second step — how the check of the approximation is done.
David Gay’s strtod() function does decimal to floating-point conversion using both IEEE double-precision floating-point arithmetic and arbitrary-precision integer arithmetic. For some inputs, a simple IEEE floating-point calculation suffices to produce the correct result; for other inputs, a combination of IEEE arithmetic and arbitrary-precision arithmetic is required. In the latter case, IEEE arithmetic is used to calculate an approximation to the correct result, which is then refined using arbitrary-precision arithmetic. In this article, I’ll describe the approximation calculation, which is based on a form of binary exponentiation.
Hexadecimal floating-point constants, also known as hexadecimal floating-point literals, are an alternative way to represent floating-point numbers in a computer program. A hexadecimal floating-point constant is shorthand for binary scientific notation, which is an abstract — yet direct — representation of a binary floating-point number. As such, hexadecimal floating-point constants have exact representations in binary floating-point, unlike decimal floating-point constants, which in general do not.
Hexadecimal floating-point constants are useful for two reasons: they bypass decimal to floating-point conversions, which are sometimes done incorrectly, and they bypass floating-point to decimal conversions which, even if done correctly, are often limited to a fixed number of decimal digits. In short, their advantage is that they allow for direct control of floating-point variables, letting you read and write their exact contents.
In this article, I’ll show you what hexadecimal floating-point constants look like, and how to use them in C.
In my article “Patterns in the Last Digits of the Positive Powers of Five” I showed that the cycles of ending digits of the positive powers of five could be represented with a binary tree:
Binary Tree Showing Nested Ending Digit Patterns of the Positive Powers of Five
The tree layout shows that certain pairs of ending digits are related, and that these pairs differ by five in their starting digits. I will show why this is true.
In my article “Counting Binary and Hexadecimal Palindromes” I derived formulas for counting binary palindromes and hexadecimal palindromes. For each type of palindrome, I derived two pairs of formulas: one pair to count n-digit palindromes, and one pair to count palindromes of n digits or less.
In this article, I will derive similar formulas to count binary/hexadecimal palindromes — multi-base palindromes I’ve shown to have an algorithmically defined structure.
To perform arbitrary-precision arithmetic in C and C++ programs on Windows, I use GMP. In particular, I use MPIR, a Windows port of GMP. MPIR is a simple alternative to using GMP under Cygwin or MinGW.
I will show you how to install MPIR in Microsoft Visual C++ as a static, 32-bit library. I will also show you how to install the optional C++ interface — also as a static, 32-bit library. I will provide two example C programs that call the GMP integer and floating-point functions, and two equivalent C++ programs — programs that use the same GMP functions, only indirectly through the C++ interface.
How many nonzero, n-digit, decimal number palindromes are there? These two formulas give the answer:
How many nonzero, decimal number palindromes are there, consisting of n-digits or less? These two formulas give the answer:
So for example, there are 900 5-digit decimal palindromes, 9,000 8-digit decimal palindromes, 1,098 decimal palindromes of 5 digits or less, and 19,998 decimal palindromes of 8 digits or less.
In this article, I will derive similar formulas to count binary and hexadecimal number palindromes.
